Effective equidistribution of orbits under semisimple groups on congruence quotients

TL;DR

This paper establishes an effective rate of equidistribution for periodic orbits of semisimple groups on congruence quotients, extending prior results by including orbits with nontrivial centralizers.

Contribution

It introduces a new effective equidistribution theorem for semisimple group orbits on congruence quotients, accommodating orbits with nontrivial centralizers.

Findings

Proves effective equidistribution for a broader class of orbits.

Utilizes an effective closing lemma in the proof.

Extends previous work by Einsiedler, Margulis, and Venkatesh.

Abstract

We prove an effective equidistribution result for periodic orbits of semisimple groups on congruence quotients of an ambient semisimple group. This extends previous work of Einsiedler, Margulis and Venkatesh. The main new feature is that we allow for periodic orbits of semisimple groups with nontrivial centralizer in the ambient group. Our proof uses crucially an effective closing lemma from work with Lindenstrauss, Margulis, Mohammadi, and Shah.